Which expressions are equivalent to [tex]-9\left(\frac{2}{3} x+1\right)[/tex]? Check all that apply.

A. [tex]-9\left(\frac{2}{3} x\right) + 9(1)[/tex]
B. [tex]-9\left(\frac{2}{3} x\right) - 9(1)[/tex]
C. [tex]-9\left(\frac{2}{3} x\right) + 1[/tex]
D. [tex]-6 x + 1[/tex]
E. [tex]-6 x + 9[/tex]
F. [tex]-6 x - 9[/tex]

Let's solve the problem by applying the distributive property and then comparing the given choices.

The original expression is:
[tex]\[ -9\left(\frac{2}{3} x + 1\right) \][/tex]

To simplify this expression, let's distribute [tex]\(-9\)[/tex] to both terms inside the parenthesis:

1. Distribute [tex]\(-9\)[/tex] to [tex]\(\frac{2}{3}x\)[/tex]:
[tex]\[
-9 \cdot \frac{2}{3}x = -\frac{18}{3}x = -6x
\][/tex]

2. Distribute [tex]\(-9\)[/tex] to [tex]\(1\)[/tex]:
[tex]\[
-9 \cdot 1 = -9
\][/tex]

So, the simplified expression is:
[tex]\[ -6x - 9 \][/tex]

Now, let's evaluate each of the given options to see if they are equivalent to [tex]\(-6x - 9\)[/tex]:

1. [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]:
[tex]\[ -6x + 9 \][/tex]
This is not equivalent to [tex]\(-6x - 9\)[/tex].

2. [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]:
[tex]\[ -6x - 9 \][/tex]
This is equivalent to the original expression.

3. [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]:
[tex]\[ -6x + 1 \][/tex]
This is not equivalent to [tex]\(-6x - 9\)[/tex].

4. [tex]\(-6 x + 1\)[/tex]:
This is not equivalent to [tex]\(-6x - 9\)[/tex].

5. [tex]\(-6 x + 9\)[/tex]:
This is not equivalent to [tex]\(-6x - 9\)[/tex].

6. [tex]\(-6 x - 9\)[/tex]:
This is equivalent to the original expression.

The expressions that are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] are:

[tex]\[ \boxed{-9\left(\frac{2}{3} x\right) - 9(1) \quad \text{and} \quad -6x - 9} \][/tex]